It is folly to try to do what you're talking about by mapping to 3D Effect, because you would need to start with a mercator projection, not an undistorted array of circles.

3D Effect does not map artwork to a sphere the way you are trying to make it do, and that will be evident in all the suggestions provided so far, including the soccer ball. See how the pentagon towards the top is "pinched" as it nears the north pole?

Think of 3D Effect as taking your rectangular artwork, wrapping it around a cylinder, and then "pinching" the top and bottom edges of it into a single point, in order to make it conform to a sphere, just like the longitude lines on the globe.

So in order to do what you want with convincing results using 3D Effect, you would need to effectively "reverse engineer" that process, "stretching" your artwork along its top and bottom edges so that it would be "unflared" when it is mapped to the sphere.

Consider this basketball which roughly simulates in principle what you would have to do to your flat array of dots:

Looks pretting convincing, right? Look again. Note how the center ridge still tapers to a single point at the visible pole.

Although disguised, there is a similar "pinching" in this golf ball, which was done by a 2D "Fisheye" envelope distortion. But it actually has a "pinching" effect at four points (because of Illustrator's inferior envelopes):

So this is one of many kinds of situations in which a regular 2D construction is more expedient than Illustrator's 3D Effect. The samples below are purely 2D constructions by the principles of orthographic projection, and don't have the polar "pinching" problem:

No Adobe at all. It was done with a plugin called Flaming Pear Flexify, V.2. Input parameter was Equilateral, and output parameter was Orthographic. I used only the image Monica provided for the end result.